Identification of Traitors in Algebraic-Geometric Traceability Codes

1. Identification of Traitors in Algebraic-Geometric Traceability Codes IEEE Trans. on Signal Processing, Vol. 52, No. 10, Oct. 2004

2. Outline • Introduction • Background • Soft-decision tracing algorithm • Tracing with additional information • Conclusions

3. Introduction • Collusion attack • Marking assumption • Upon finding a pirate copy, the goal of the distributor is to identify as many traitors as possible • A fingerprinting code C is a set of codewords with the identification property:Let z be a word that is generated by a coalition of codewords ; then, there is a algorithm that, given z, allows the recovery of at least on codeword in U

4. Background -Traceability codes • Notation

5. Background -Traceability codes • Undetectable • Descendant • c-traceability code

6. Background -Traceability codes • Theorem 1Let C be a [n,k,d]-code, if d>n(1-1/c2); then, C is a c-traceability code. • Theorem 2Let C be a [n,k,d]-code, if d>n(1-1/c2)+s/c2; then, C is a c-traceability code tolerating s erasures.

7. Background –AG codes • AG codes (geometric Goppa code) • Let X be an absolutely irreducible curve over Fq of genus g. • P1,…,Pn are Fq-rational points of X. • D=P1+…+Pn. • G is a divisor of X of degree deg(G)<n such that supp(G)∩supp(D)=Φ • L(G) denotes the linear space of G. • An AG code

8. Background-Soft-decision list decoding of AG codes • The received codeword is usually a corrupted version because of the noise on the communication channel • Solutions: • List decoding • Soft-decision list decoding

9. Background-Soft-decision list decoding of AG codes • Theorem 3Let C be a q-ary AG code of blocklength n and designed minimum distance d*=n-deg(G), and let ε>0 be an arbitrary constant. For 1≤i≤n and , let be a non-negative real. Then, there exists a deterministic algorithm with runtime poly(n,q,1/ε) that, when given as input the weights for 1≤i≤n and , finds a list of all codewords c=(c1,…,cn) of C that satisfy (1)

10. Background-Soft-decision list decoding of AG codes • A q-ary symmetric erasure channel with error probability , erasure probability , input alphabet X=Fq, and output alphabet Y=FqU{*} can be characterized by a transition probability matrix :

11. Background-Soft-decision list decoding of AG codes • Koetter & Vardy show how to improve the performance of the GS soft-decision algorithm for the q-ary symmetric erasure channel • In this paper, they define to be the reliability weights in the tracing algorithm

12. Background-Soft-decision list decoding of AG codes • If v contains (n-m) erasures and (m-l) errors, then we have that the left-hand side is maximized for

13. Background-Soft-decision list decoding of AG codes (2)  If upon receiving a word v, (n-m) symbols are erased, then for every value of l that satisfies (2), the soft-decoding algorithm will output codeword u. Therefore, the algorithm can handle (n-m) erasures and (m-l) errors.

14. Soft-decision tracing algorithm • For a AG c-traceability code tolerating s erasures with parameters [n,k,d], the goal of a tracing algorithm is to output a c-bounded list that contains all parents of a given descendant • Positive parentA codeword that agrees in at least c(k+g-1)+1 of the nonerased positions with the descendant

15. Soft-decision tracing algorithm • Theorem 4Let C be an AG c-traceability code tolerating s erasures with parameters [n,k,d]. Given a descendant, a positive parent is a provably identifiable member of the coalition of codewords that created the descendant.

16. Soft-decision tracing algorithm • Corollary 1Let C be an AG c-traceability code tolerating s erasures, with parameters [n,k,d]. Let z be a descendant of some coalition. Suppose that there are s symbols erased in z. In addition, suppose that j already-identified positive parents (j<c) jointly match less than n-s-(c-j)(k+g-1) positions of z. Then, any codeword that agrees with z in at least (c-j)(k+g-1)+1 of the unmatched positions is also a positive parent.

17. Soft-decision tracing algorithm • Given a codeword v and a descendant z, the set M(v,z)={i:vi=zi} is called the set of matching positions • When a positive parent u is identified, tracing the remaining positive parents requires searching for codewords that match the descendant in a suitable number of positions outside M(u,z)

18. Soft-decision tracing algorithm • Input: • Descendant zdescc*(Ct;s), |Ct|≤c • c,s • C: AG code with d>n(1-1/c2)+s/c2 • Output: • A list L of all positive parents of z

19. Soft-decision tracing algorithm

20. Soft-decision tracing algorithm • Correctness of the algorithm

21. Soft-decision tracing algorithm • Example • [31,4,28] RS code • 3-traceability code • 1048576 codewords

22. Soft-decision tracing algorithm • i=1

23. Soft-decision tracing algorithm • i=2

24. Soft-decision tracing algorithm • i=3

25. Tracing with additional information • Knowledge of the coalition size • Knowledge of the parents operation mode

26. Conclusions • Silverberg et al. show that tracing algorithms can be designed using hard-decision list decoding techniques. Their approach guarantees to find at least one of the parents. • This paper extends the results of Silverberg et al. • Using soft-decision decoding techniques • Considering traceability codes tolerating erasures • Allowing the search for parents whose identification depends on the previously found parents